Presumably either taxable income includes something other than "pay" (e.g. benefits, bonus etc) or "gross pay" is something of an artificial figure (e.g. like a notional salary where there is salary sacrifice for certain other benefits)

A tax code (including a K-code) only determines how much tax an employer has to deduct from your salary under PAYE.

It doesn't affect your salary/taxable income from that employment, albeit that in arriving at a K-code amount, HMRC would include those other liabilities

A tax code does not determine how much tax is to be deducted. It simply determines the amount of taxable pay. An ordinary code results in a reduction to the gross pay to arrive at the taxable pay. A K Code results in an increase to the gross pay to arrive at the taxable pay. So, looking at a PAYE calculation, the taxable pay would be more than the gross pay.

An ordinary code results in a reduction to the gross pay to arrive at the taxable pay. A K Code results in an increase to the gross pay to arrive at the taxable pay.

I would disagree with this statement (unless i am misunderstanding the context it is posted in - apologies if this is the case)

A tax code change never normally changes taxable pay amount as shown on the payslip - a tax code change only alters the "fee pay" column of the P11 calculations (and thereafter the tax deducted) - this bit is normally hidden behind the scenes and wont show up on the payslip so a tax code change will not be the reason for the difference between Gross Pay and Taxable Pay - unless something non standard is going on.